Namespaces
Variants
Actions

Real-analytic space

From Encyclopedia of Mathematics
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.


An analytic space over the field $ \mathbf R $ of real numbers. Unlike in the case of complex-analytic spaces, the structure sheaves of real-analytic spaces need not be coherent (cf. Coherent sheaf). Real-analytic spaces are said to be coherent if their structure sheaves are coherent. All real-analytic manifolds (i.e. smooth real-analytic spaces) are coherent real-analytic spaces.

Let $ V _ {a} $ be the germ at a point $ a $ of a real-analytic subset of $ \mathbf R ^ {n} $ (cf. Analytic set). This defines the germ at $ a $ of a complex-analytic subset $ \widetilde{V} _ {a} $ of the space $ \mathbf C ^ {n} $ with the following equivalent properties: 1) $ \widetilde{V}_ {a} $ is the intersection of all germs of complex-analytic sets containing $ V _ {a} $; 2) if $ {\mathcal O} _ {V _ {a} } $ is the analytic algebra of the germ $ V _ {a} $, then $ {\mathcal O} _ {V _ {a} } \otimes \mathbf C $ is the analytic algebra of the germ $ \widetilde{V}_ {a} $. The germ $ \widetilde{V}_ {a} $ is said to be the complexification of the germ $ V _ {a} $, and $ V _ {a} $ is said to be the real part of the germ $ \widetilde{V}_ {a} $. Analogously, for any coherent real-analytic countably-infinite space $ X $ it is possible to construct the complexification $ \widetilde{X} $, which is a complex-analytic space. $ X $ will then have a fundamental system of neighbourhoods in $ \widetilde{X} $ which are Stein spaces (cf. Stein space).

The theory of coherent real-analytic spaces is similar to the theory of complex Stein spaces. Global sections of any coherent analytic sheaf of modules $ F $ on a coherent real-analytic countably-infinite space $ X $ generate modules of germs of its sections at any point of $ X $, and all groups $ H ^ {q} ( X, F ) $ vanish if $ q \geq 1 $.

For any finite-dimensional coherent real-analytic countably-infinite space $ ( X, {\mathcal O} _ {X} ) $ there exists a morphism

$$ f = ( f _ {0} , f _ {1} ): ( X, {\mathcal O} _ {X} ) \rightarrow \ ( \mathbf R ^ {n} , {\mathcal O} _ {\mathbf R ^ {n} } ) $$

such that $ f _ {0} $ is a proper one-to-one mapping of $ X $ into a coherent subspace in $ \mathbf R ^ {n} $, while $ f $ is an imbedding at the smooth points of $ X $. In particular, any (Hausdorff and countably-infinite) real-analytic manifold is isomorphic to a real-analytic submanifold in $ \mathbf R ^ {n} $. For a reduced coherent real-analytic space $ X $ the set of classes of isomorphic real-analytic principal fibre bundles with a real structure Lie group $ G $, admitting complexification, and base $ X $ is in one-to-one correspondence with the set of classes of isomorphic topological principal fibre bundles with the same structure group $ G $.

References

[1] A. Tognoli, "Some results in the theory of real analytic spaces" M. Jurchesan (ed.) , Espaces Analytiques (Bucharest 1969) , Acad. Roumanie (1971) pp. 149–157

Comments

References

[a1] H. Cartan, "Variétés analytiques réelles et variétés analytiques complexes" Bull. Soc. Math. France , 85 (1957) pp. 77–99
[a2] F. Bruhat, H. Cartan, "Sur la structure des sous-ensembles analytiques réels" C.R. Acad. Sci. Paris , 244 (1957) pp. 988–900
[a3] F. Bruhat, H. Cartan, "Sur les composantes irréductibles d'un sous-ensemble" C.R. Acad. Sci. Paris , 244 (1957) pp. 1123–1126
[a4] F. Bruhat, H. Whitney, "Quelques propriétés fondamentales des ensembles analytiques-réels" Comm. Math. Helv. , 33 (1959) pp. 132–160
[a5] R. Narasimhan, "Introduction to the theory of analytic spaces" , Lect. notes in math. , 25 , Springer (1966)
[a6] H. Grauert, R. Remmert, "Theory of Stein spaces" , Springer (1979) (Translated from German)
How to Cite This Entry:
Real-analytic space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Real-analytic_space&oldid=52048
This article was adapted from an original article by D.A. Ponomarev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article